¼ ui Á i :

L(ui ) ¼ x dF(x) ¼ (2:31)

m m m

ui

0 0

^

Natural estimates of these quantities are computed as ui ¼ Xri :n , where ri ¼ bnui c, and

Pri

j¼1 Xj:n ^

g

^ % ui i ,

L(ui ) ¼ Pn

^

m

j¼1 Xj:n

P

with gi ¼ ri Xj:n =ri and m ¼ X n . This shows that the asymptotics of sample

^ ^

j¼1

Lorenz curve ordinates can be reduced to the asymptotics of the sample quantiles for

which there is a well-developed theory (e.g., David, 1981, pp. 254 “ 258).

Under the assumption that the observations form a simple random sample from an

underlying distribution with ¬nite variance (s 2 , say), Beach and Davidson showed

^

that the (k þ 1)-dimensional random vector u :¼ (u1 g1 , . . . , uk gk , ukþ1 gkþ1 )`, with

^ ^ ^

^

ukþ1 ¼ 1 and gkþ1 ¼ X n, is asymptotically jointly multivariate normal

p¬¬¬ d

^

n(u À u) À N (0, V), (2:32)

!

34 GENERAL PRINCIPLES

where the asymptotic covariance matrix V ¼ (vij ) is given by

vij ¼ ui l2 þ (1 À ui ){F À1 (ui ) À gi }{F À1 (uj ) À gj }

i

!

þ {F À1 (ui ) À gi }(gj À gi ) , i j: (2:33)

Here l2 :¼ var[X jX F À1 (ui )]. (Note that vkþ1,kþ1 =n ¼ s 2 =n, the variance of the

i

sample mean.) Writing

^ ^ ^

L(u) :¼ [L(u1 ), . . . , L(uk )]` ¼ [u1 g1 =(ukþ1 gkþ1 ), . . . , uk gk =(ukþ1 gkþ1 )]`

^ ^ ^ ^

an application of the delta method shows that the k-dimensional vector of sample

^

Lorenz curve ordinates L(u) is also jointly multivariate normal, speci¬cally

p¬¬¬ d

^

n[L(u) À L(u)] À N (0, V ), (2:34)

!

where V ¼ (vij ) is given by

ui gi uj gj

1

s2

vij ¼ vij þ

m2 m2 m2

uj gj

ui gi

vj,kþ1 À vi,kþ1 , i j: (2:35)

À

m3 m3

We see that V depends solely on the ui , the unconditional mean and variance m

and s 2, the income quantiles F À1 (ui ), and the conditional means and variances gi

and l2 . All these quantities can be estimated consistently. For example, a natural

i

consistent estimator of l2 is

i

1X ri

^ (Xj:n À gi )2 :

li2 ^ (2:36)

¼

ri j¼1

Consequently, tests of joint hypotheses on Lorenz curve ordinates are now

available in a straightforward manner. For example, if it is required to compare an

^ ^ ^

estimated Lorenz curve L(u) ¼ [L(u1 ), . . . , L(uk )] against a theoretical Lorenz curve

L (u) ¼ [L (u1 ), . . . , L (uk )] in order to test H0 : L(u) ¼ L0 (u), we may use the

0 0 0

quadratic form

^ ^ ^

n[L(u) À L0 (u)]` V À1 [L(u) À L0 (u)],

a statistic that is asymptotically distributed as a xk2 under the null hypothesis.

Similarly, to compare two estimated Lorenz curves from independent samples in

order to test H0 : L1 (u) ¼ L2 (u), we may use

^ ^ ^ ^ ^ ^

n[L1 (u) À L2 (u)]` [V1 =n1 þ V2 =n2 ]À1 [L1 (u) À L2 (u)],

which is also asymptotically distributed as a xk2 under the null hypothesis.

35

2.1 SOME CONCEPTS FROM ECONOMICS

These procedures are omnibus tests in that they have power against a wide variety

of differences between two Lorenz curves. In particular, they are asymptotically

distribution-free and consistent tests against the alternative of crossing Lorenz curves.

This line of research has been extended in several directions: Beach and

Richmond (1985) provided asymptotically distribution-free simultaneous con¬dence

intervals for Lorenz curve ordinates. More recently, Dardanoni and Forcina (1999)

considered comparisons among more than two population distributions using

methodology from the literature on order-restricted statistical inference. Davidson

and Duclos (2000) derived asymptotic distributions for the ranking of distributions

in terms of poverty, inequality, and stochastic dominance of arbitrary order (note that

the generalized Lorenz order is equivalent to second-order stochastic dominance).

Zheng (2002) extended the Beach “Davidson results to strati¬ed, cluster, and

multistage samples.

Inequality Measures

Several possibilities exist to study the asymptotic properties of inequality measures.

Since the classical and generalized Gini coef¬cients are de¬ned in terms of the Lorenz

curve, a natural line of attack is to consider them as functionals of the empirical

Lorenz process. This was the approach followed by Goldie (1977) for the case of the

classical Gini coef¬cient and by Barrett and Donald (2000) for the generalized Gini.

However, an alternative approach using somewhat simpler probabilistic tools (not

requiring empirical process techniques) is also feasible. It was developed by Sendler

(1979) for the classical Gini coef¬cient and more recently by Zitikis and Gastwirth

(2002) for the generalized version. An additional bene¬t is that fewer assumptions”

essentially only moment assumptions”are required.

Using general representations for the moments of order statistics (see David,

1981), the generalized Gini coef¬cient (2.24) can be expressed in the form

° °

n 1 À1 1 1 À1

nÀ1

F (t) d{(1 À t)n }:

Gn ¼ 1 À F (t)(1 À t) dt ¼ 1 þ (2:37)

m0 m0

This suggests that Gn can be estimated by

1 XÈ n É

^ (n À i þ 1)n À (n À i)n Xi:n ,

Gn,n ¼1À

X n nn i¼1